FUZZY COMPREHENSIVE EVALUATION METHOD FOR SYMMETRY DEGREE OF MECHANICAL STRUCTURE SYMMETRY. Fan Liu and Zhiyong Ma

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1 FUZZY COMPREHENSIVE EVALUATION METHOD FOR SYMMETRY DEGREE OF MECHANICAL STRUCTURE SYMMETRY Fan Liu and Zhiyong Ma Faculty of Mechanical Engineering and Mechanics, Zhejiang Provincial Key Lab of Part Rolling Technology, Ningbo University, Ningbo Zhejiang, China Received June 206, Accepted May 207 No. 6-CSME-8, E.I.C. Accession 3967 ABSTRACT This paper proposes a fuzzy comprehensive evaluation method to determine the symmetry degree of three types of mechanical structure symmetries. In the proposed method, factors are set based on four symmetry elements; the analytic hierarchy process is used to determine the weight set, and an experiment was developed to determine the weight values. Based on the maximum membership rule, the type of symmetry of the evaluation object can be determined based on the evaluation result. Finally, instances were applied to illustrate the proposed evaluation method, which validated the proposed method is successful. Keywords: mechanical structure; structure symmetry; symmetry degree; symmetry element; fuzzy comprehensive evaluation. MÉTHODE GLOBALE FLOUE POUR L ÉVALUATION DU DEGRÉ DES SYMÉTRIES D UNE STRUCTURE MÉCANIQUE RÉSUMÉ L article propose une méthode globale floue pour l évaluation du degré des symétries d une structure mécanique. Dans la méthode proposée les facteurs sont déterminés sur la base de quatre éléments de symétrie ; le processus d analyse hiérarchique est utilisé pour déterminer les paramètres du poids étalon, et une expérimentation a été développée pour déterminer les valeurs pondérales. En se basant sur la règle d adhésion maximale, le type de symétrie de l objet évalué peut être déterminé selon le résultat de l évaluation. Finalement des cas d exemples ont été appliqués pour illustrer la méthode d évaluation proposée. Ces exemples ont validé le succès de la méthode. Mots-clés : structure mécanique; symétrie d une structure; élément de symétrie; évaluation globale floue. Transactions of the Canadian Society for Mechanical Engineering, Vol. 4, No. 3,

2 Fig.. Three types of gears. Fig. 2. Two box covers.. INTRODUCTION Symmetry wildly exists in natural and artificial systems. For example, hands of human beings are mirror symmetric, snowflakes are rotation symmetric, wheels of vehicles are translation symmetric, and sunrise and sunset are time symmetric. In many branches of science, such as natural science [ 3], social science [4], and engineering [5], researchers focused on symmetry and achieved good results [6, 7]. Several symmetric forms exist in mechanical products such as function symmetry, principle symmetry, structure symmetry, and technical symmetry [8]. Among them, structure symmetry is the most familiar. If a mechanical system has two or more identical geometric structures arranged in an orderly manner according to some rule or regulation, it is said to have structure symmetry [9, 0]. Based on the degree of symmetry, mechanical structures can be divided into three types of symmetries such as strict, broken, and weak structure symmetries. Fig. shows three types of gears having rotation symmetry. However, they have different distributions of gear teeth. Based on the distribution of gear teeth, gears (a), (b), and (c) have strict, broken, and weak structure symmetries, respectively. For the two box covers shown in Fig. 2, cover (a) has strict-structure symmetry, and the position of fixing hole (2) in cover (b) is different from that of cover (a); moreover, cover (b) has broken-structure symmetry. For the chain plates shown in Fig. 3, plate (a) has strict structure symmetry, and the structure to the right side of the axis of plate (b) is not completely similar to that of the left; hence, plate (b) has broken-structure symmetry. Symmetry means the invariance with respect to a transformation group in the physical world. In contrast, broken symmetry reflects the differences between the transformations. Broken symmetry in mechanical structures reflects the differences between multiple repeated components of mechanical systems. The greater the difference, lower the degree of similarity and repeatability and higher the degree of broken symmetry. Structure symmetry plays important roles in realizing functions, improving performances, and satisfying constraints of mechanical products. Although the three types of symmetries aforementioned share similar functions in mechanical systems, their functionalities change under different degrees of symmetry. In general, for a special symmetric structure, if the degree of symmetry is altered, its function will also change. In Fig. 2, the symmetry degree of cover (b) is less than that of cover (a), and the asymmetric fixing holes of cover (b) are designed to improve the positioning accuracy. In Fig. 3, the symmetry degree of plate (b) is 338 Transactions of the Canadian Society for Mechanical Engineering, Vol. 4, No. 3, 207

3 Fig. 3. Two chain plates. less than that of plate (a), and the irregular structure on the right side of cover (b) reduces the impact force and improves the stability of transmission. Owing to the significance of symmetry in mechanical structures, it is important to systematically research the application laws and different functions of the three types of structure symmetries, the change in functions due to the change in structure when strict structure symmetry is transformed into broken and weak structure symmetries, and the application methods. In the aforementioned studies, determining the type of symmetry of a symmetric structure is an important, basic issue. Although the three types of structure symmetries have symmetric characteristics, their symmetry degrees are different. By calculating the symmetry degree, the type of symmetry can be determined. Shao et al. [] divided a symmetric structure into characteristic points on its surface, and proposed a method to determine whether a structure is symmetric by matching the characteristic points. Zheng et al. [2] established a two-stage algorithm to detect part symmetry by comparing boundary boxes of rotated parts to compute the symmetry boundary. Tate [3] proposed a pattern-matching algorithm to calculate the symmetry degree by comparing and matching the surfaces of products. Liu et al. [4] employed dynamic characteristic curves to imitate the changes in symmetry main body and determine the symmetry degree by using overlap ratio of curves before and after performing symmetry operation. Other methods that detect symmetry and symmetry degree were analyzed in the studies by Ma [5], Lee [6], and Bona [7]. Existing algorithms detect symmetry and calculate the degree of symmetry to a reasonable degree, but they still have some shortcomings: () the algorithms are complex and ineffective; (2) the algorithms cannot cover all types of structure symmetries, particularly complex structure symmetries such as rotation translation symmetry and translation mirror symmetry; (3) some algorithms only detect symmetry and cannot calculate symmetry degree. In this study, we propose a fuzzy comprehensive evaluation (FCE) method to calculate the symmetry degree of symmetric structures. Owing to the complexity and variety of symmetric structures, it is hard to calculate the symmetry degree accurately, based on geometric structures. By using the FCE method, the symmetry degree of all types of structure symmetries can be evaluated, and symmetry elements, such as symmetry main body, symmetry component, symmetry standard, and symmetry operation, can be analyzed. Symmetry elements affect the degree of symmetry to different extents. The FCE method can calculate detailed quantifiable degree of symmetry based on all the elements and their different effects. Finally, an experiment was designed to ascertain the weight vectors of the factors. Based on the maximum membership rule, the type of symmetry can be determined after obtaining the evaluation result. 2. FUZZY COMPREHENSIVE EVALUATION METHOD The FCE method, first proposed by Zadeh in 965, is a mathematical method to evaluate objects affected by various factors [8 20]. The FCE method gives an overall evaluation of objects by using fuzzy mathematics to transform qualitative evaluation into quantitative evaluation. Figure 4 shows the main steps of the FCE method. In this study, the evaluation object is the degree of symmetry. The factors of the factors set U are Transactions of the Canadian Society for Mechanical Engineering, Vol. 4, No. 3,

4 Fig. 4. The main steps of FCE method. Fig. 5. A rhomboid link mechanism. set based on all the symmetry elements, and the evaluation standards in the evaluation set V correspond to the three types of structure symmetries. The single-factor evaluation matrix R from U to V is set based on the influences of symmetry elements on the degree of symmetry. The weight vectors (W) of the factors is developed based on the analytic hierarchy process (AHP) method. 3. ELEMENTS OF MECHANICAL STRUCTURE SYMMETRY In mechanical systems, many different types of structure symmetries exist. The symmetry type and symmetry degree of a symmetric structure are influenced by various symmetry elements. Mechanical symmetry has four basic elements, such as symmetry main body, symmetry component, symmetry standard, and symmetry operation. The symmetry main body is the element that supports the symmetric structure. The symmetry components are similar structures that repeat regularly. The symmetry standard means the repeating standard of symmetry components. The symmetry operation means the repeating form of symmetry components. Figure 5 shows a rhomboid link mechanism, which is a typical mirror symmetric structure. In the rhomboid link mechanism, the symmetry main body is the frame (9), the symmetry components are 340 Transactions of the Canadian Society for Mechanical Engineering, Vol. 4, No. 3, 207

5 links () to (8), the symmetry standards are lines A and B, and the symmetry operations are two mirror operations. The elements of structure symmetry have many specific attributes. Every symmetric structure has all the four symmetry elements. The differences in the specific attributes of the four symmetry elements lead to specific symmetry type and symmetry degree. Table lists the attributes of the symmetry elements and gives their schematics. 4. FCE FOR SYMMETRY DEGREE The structure of mechanical structure symmetries is multivariate. Based on the types of structures, structure symmetry can be divided into three basic, five combinational, and six scaling symmetries. Based on the symmetry degree, structure symmetry can be divided into strict, broken, and weak. The type of symmetry and symmetry degree depend on the specific types of symmetry elements and their attributes. The symmetry degree of structure symmetries is not only influenced by geometric structure, but also by other factors, making it difficult to develop a simplex geometric method to evaluate symmetry degree. The evaluation method must be suitable for all types of structure symmetries and relate to all influential factors. Many of the influential factors cannot be qualitatively evaluated. Based on the above analysis, the FCE is a good solution to evaluate the symmetry degree of structure symmetries. The FCE algorithm developed for evaluating the symmetry degree is explained in Sections The Factors Set The factors of the factors set U are the elements that have influences on structure symmetry. The objective of establishing a method that determines the degree of symmetry is to understand the different functions arising from the different degrees of symmetries. In mechanical systems, the functions realized by different degrees of symmetries can be distinguished. Owing to the complexity of the realization principle between a symmetric structure and its function, we need to analyze not only the geometry, but also the movement of components to understand the realization principle. Hence, to evaluate the symmetry degree of mechanical structure symmetries, we need to consider all the symmetry elements systematically, including the symmetry body, symmetry component, symmetry standard, and symmetry operation. Each attribute of the four symmetry elements has a special influence on the formation of structure symmetry [2]. The factors set U is established based on the nine specific attributes of the four symmetry elements, as shown in Eq. (). Table 2 lists the nine factors of the factors set U. U = {u, u 2, u 3, u 4, u 5, u 6, u 7, u 8, u 9 }. () 4.2. The Evaluation Set The evaluation set V includes three evaluation standards, as seen from Eq. (2). In the evaluation set, v, v 2, and v 3 correspond to strict, broken, and weak structure symmetries, respectively. V = {v, v 2, v 3 }. (2) 4.3. The Single-Factor Evaluation Matrix The single-factor evaluation matrix R is the evaluation from U to V, which shows the level of influence of each symmetry element (u i ) on the symmetry degree (V ). The single-factor evaluation matrix R can be expressed using Eq. (3). In Eq. (3), R i (r i, r i2, and r i3 ) is the evaluation from single factor u i to V (v, v 2, and v 3 ). For every R i, the evaluations must meet the normalized conditions, and the sum of the values of the Transactions of the Canadian Society for Mechanical Engineering, Vol. 4, No. 3,

6 Table. Attributes of symmetry elements. 342 Transactions of the Canadian Society for Mechanical Engineering, Vol. 4, No. 3, 207

7 Table 2. The factors of the factors set U. vectors is, i.e., r i + r i2 + r i3 =. R = R R 2. R 9 = r r 2 r 3 r 2 r 22 r r 9 r 92 r 93. (3) Due to the complexity and variety of symmetric structures, it is hard to determine the level of influence of symmetry elements on the symmetry degree. The basic qualitative descriptions of the evaluations from U to V are given in Table 3. The influences of the symmetry elements on the symmetry degrees vary depending on the various symmetric structures. The exact evaluation value from u i to v i need to be calculated based on the particular structure. In previous studies, we collected a large number of instances of mechanical structure symmetry. Based on the study on instances, we analyzed the importance of the influence of each detailed attributes in the four symmetry elements and proposed detailed attributes of every factor ui. Table 3 gives the suggested value range of the evaluation from u i to v i. The definitions of the parameters given in Table 3 are explained below:. u : the regularity of structure of the symmetry main body. The structure of the main body is divided into two types such as regular shape and irregular shape. The more regular the shape of the main body, the higher the symmetry degree. Among the regular shapes, the spheroid main body has the highest symmetry degree, followed by the cylinder and cuboid, and then the pyramid and cone. 2. u 2 : the quantity of symmetry components. The symmetry degree of the continuous components is higher than that of the discrete components. For the discrete components, the more the quantity of components, the higher the symmetry degree. Transactions of the Canadian Society for Mechanical Engineering, Vol. 4, No. 3,

8 Table 3. The suggested evaluation from U to V. 3. u 3 : the similarity of symmetry components. The similarity of the symmetry components is divided into three regions: high, medium, and low similarities. The high similarity components have the highest symmetry degree, followed by the medium and low similarity components. 4. u 4 : the directionality of symmetry components. Based on the directionality, the symmetry component can be divided into bidirectional and unidirectional components. The symmetry degree of the bidirectional components is higher than that of the unidirectional components. 5. u 5 : the distribution of the symmetry components. Based on the distribution, the symmetry components can be divided into completely distributed and partially distributed components. The symmetry degree of the completely distributed components is higher than that of the partially distributed components. 6. u 6 : the quantity of symmetry standards. The more the quantity of symmetry standards, the higher the symmetry degree. 7. u 7 : the overlap of symmetry standards. The overlap of the symmetry standards is divided into three layers, such as high overlap, medium overlap, and low overlap. The higher the overlap of symmetry standards, the higher the symmetry degree. 8. u 8 : the regularity of the symmetry operation. The regularity of the symmetry operation is divided into three regions, such as high, medium, and low regularities. The higher the regularity of the symmetry operation, the higher will be the symmetry degree. 9. u 9 : the combination of the symmetry operation. Based on the combination, the symmetry operation can be divided into two types, such as simplex and combinational operations, and the combinational operation can be divided into un-scaling and scaling. In these operations, the un-scaling combinational operation has the highest symmetry degree, followed by the scaling combinational operation, and then the simplex operation. It must be pointed out that the determination of the evaluation r i j from u i to v i is largely a subjective process. The suggested range of values, shown in Table 3, is suitable for most symmetric structures. For a 344 Transactions of the Canadian Society for Mechanical Engineering, Vol. 4, No. 3, 207

9 Table 4. The relative importance. Degree The relative importance d i j between u i and u j Equally important 3 Generally more important 5 Far more important 7 More important at the second highest degree 9 More important at the first highest degree 2, 4, 6, and 8 For compromises between the above special structure, a good solution would be to obtain evaluations from multiple domain experts and select the average as the final evaluation value The Weight Vectors of the Factors The influences of the factors u i (i = 9) on the symmetry degree are different. To describe the influences of u i, the weight vectors W of the factors need to be developed, as seen in Eq. (4). In Eq. (4), w i are the values of the influences the factors u i have on the symmetry degree. W = {w, w 2, w 3, w 4, w 5, w 6, w 7, w 8, w 9 }. (4) There are many methods to establish the weight vectors of factors. In this study, we employ the AHP method to develop the weight vectors. The AHP, developed by Saaty [22] in the 970s, is a structured technique for analyzing complex decisions. The process of establishing the weight vectors W is as follows: () Compare the factors in the factors set U individually, sort the factors according to the importance of the influence on the symmetry degree. Establish the comparison matrix D to express the relative importance between the factors. The comparison matrix D is shown in Eq. (5). In Eq. (5), d i j (i = 9; j = 9) is used to describe the relative importance between u i and u j, d i j = /d i j, when i = j, d i j =. The value of the relative importance can be determined based on Table 4. Due to the complexity of mechanical structures, it is hard to determine the precise value of the relative importance. Based on the process and the roles of the factors in the generation of mechanical structure symmetry, we analyzed the relative importance of the factors and developed a qualitative description, as seen in Fig. 6. We will introduce the precise value of the relative importance in the next section. D = d d 2... d 9 d 2 d d d 9 d d 99. (5) (2) Calculate the eigenvector W and the maximum eigenvalue λ max of the comparison matrix; the eigenvector W is the weight vector of the factor. (3) Perform the normalization process for the comparison matrix. Calculate the consistency degree CI using Eq. (6) and the random consistency ratio CR using Eq. (7). In Eqs. (6) and (7), n is the number of the comparison matrix, and RI is the average random consistency index. The comparison matrix D and the weight vectors W can be used only when CR < 0.; otherwise, the comparison matrix D needs to be developed again. CI = λ max n n, (6) CR = CI RI. (7) Transactions of the Canadian Society for Mechanical Engineering, Vol. 4, No. 3,

10 Fig. 6. The relative importance of the factors The Comprehensive Evaluation By multiplying the weight vectors W and the single-factor evaluation matrix R, the comprehensive evaluation B can be obtained using Eq. (8). In Eq. (8), b i means the membership degree of evaluation object to the evaluation standard v i. To make evaluation B more intuitional, we performed the normalization process for the evaluation B by using Eq. (9) and obtained the normalized evaluation B. The final evaluation result is determined based on the maximum membership rule, i.e., the evaluation standard v i will be the evaluation result, if b i is maximum, as shown in Eq. (0). Based on the evaluation standard v i, the type of symmetry (strict, broken, or weak structure symmetries) of the evaluation object can be established. B = W R = (w, w 2,..., w 9 ) R R 2. R 9 = (b, b 2, b 3 ) (8) B = (b, b 2, b 3), b i = b i 3 i= b, i =,2,3 (9) i v = {v i /v i max(b i)}, i =,2,3 (0) 5. EXPERIMENT FOR WEIGHT VECTORS OF THE FACTORS In the previous section, we proposed a qualitative description of relative importance of the factors, but the precise value of the relative importance needs to be determined. The value of the relative importance can be determined by domain experts. Multiple domain experts propose the value of relative importance separately, and the final value can be determined by calculating the average of the proposed values. The drawback of this method is that the value is effected by the subjectivity of experts to a significant extent. In this study, we designed an experiment to determine the precise value of the relative importance. In the experiment, the value is solved based on symmetric instances; hence, the value is more objective than that proposed by experts. 346 Transactions of the Canadian Society for Mechanical Engineering, Vol. 4, No. 3, 207

11 Fig. 7. The experimental process. Table 5. The kinds of symmetries and types of instances under them. Kind of Translation Rotation Mirror Slide Reverse Rotation- Other symmetry symmetry symmetry symmetry symmetry symmetry translation symmetries symmetry Strict symmetry Symmetry broken Weak symmetry Experiment Design We designed an experiment to determine the precise value of the relative importance, shown in Fig. 7. The experiment process is as follows: Step : Collect symmetric instances: we collected 60 symmetric instances to verify the rationality of the value of relative importance. To have an accurate experiment result, we need to include the three types of symmetries and various instances under each symmetry type. Table 5 lists the kinds of symmetries and types of instances under each symmetry. Step 2: Set the single-factor evaluation matrix of the instances: Based on the suggested range of values shown in the above section, analyze the structure and set the single-factor evaluation matrix of the instances. Step 3: Set the relative importance of the comparison matrix: By referring to the qualitative description of the relative importance, shown in Fig. 6, set an initial value of the relative importance and calculate the weight vectors of the factors. Transactions of the Canadian Society for Mechanical Engineering, Vol. 4, No. 3,

12 Fig. 8. The interface of setting the relative importance. Fig. 9. The interface of setting information of instances. Step 4: Comprehensive evaluation: Based on the value of the single-factor evaluation matrix and the weight vectors calculated in Steps 2 and 3, calculate the comprehensive evaluation and obtain the evaluation results. Compare the evaluation results with the actual symmetry degrees of the instances. If the evaluation results are same as the symmetry degrees of instances, the experiment ends, and the weight vector of the factors set 348 Transactions of the Canadian Society for Mechanical Engineering, Vol. 4, No. 3, 207

13 in Step 3 is the experiment result. If any of the evaluation result is not the same as the symmetry degrees of instances, repeat Steps 2, 3, and 4 until all the evaluation results are same as the symmetry degrees of the instances. We developed a software to conduct the experiment. Figure 8 shows the interface of setting the relative importance and calculating the weight vectors. Figure 9 shows the interface of setting the information of the instances and calculating the evaluation results Experiment Result Based on the above steps of the experiment, we obtained the value of the relative importance and the weight vectors that best fit the symmetric instances. The experiment results are expressed in Eq. () and Fig. 0: D = () Fig. 0. The weights of factors. 6. APPLICATION In this section, we apply two symmetric structures to illustrate the proposed evaluation method. Figure shows the structure of a bicycle expansion brake. The brake includes four primary parts, such as the brake block, the brake block 2, the brake cam 3, the shell 4, and the spring. The brake has strict mirror symmetry, and the primary mirror-symmetric parts are the two brake blocks. Based on the proposed evaluation method, we evaluate the symmetry degree of the brake as follows. () Analyze the factors that influence the symmetry degree. The following are the descriptions of the nine factors that influence the symmetry degree: u the structure of the shell; u 2 the quantity of the brake blocks; u 3 the similarity of the structure of the brake blocks; u 4 the directionality of the brake blocks; Transactions of the Canadian Society for Mechanical Engineering, Vol. 4, No. 3,

14 Fig.. A bicycle expansion brake. Fig. 2. The improved bicycle expansion brake. u5 the distribution of the brake blocks; u6 the quantity of the standard of the mirror symmetry; u7 the overlap of the standard of the mirror symmetry; u8 the regularity of the mirror operation; u9 the combination of the brake blocks. (2) Establish the single-factor evaluation matrix based on the characteristics of the structure, as given in Table 6. (3) Calculate the comprehensive evaluation. Based on the weight vectors of the factors proposed in the experiment, the comprehensive evaluation was calculated, B0 = (b0, b02, b03 ) = (0.6965, 0.53, 0.504). According to the maximum membership rule, the evaluation result is v (strict-structure symmetry), the same as the actual symmetry degree of the brake. Figure 2 shows an improved bicycle expansion brake. Different from the brake shown in Fig., the number of blocks in the improved brake is changed from two to one, and the structure of the brake block has broken, mirror symmetry. For the comprehensive evaluation of the improved brake, the factors are same as that of the original brake, but the single-factor evaluation matrix is different, given in Table 7. The comprehensive evaluation of the improved brake is B0 = (b0, b02, b03 ) = (0.356, , ), and the 350 Transactions of the Canadian Society for Mechanical Engineering, Vol. 4, No. 3, 207

15 Table 6. The single-factor evaluation matrix elements of the brake. u u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 v v v Table 7. The single-factor evaluation matrix elements of the improved brake. u u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 v v v Table 8. The single-factor evaluation matrix elements of normal gear mechanism and intermittent gear mechanism. evaluation result is v 2 (structure symmetry broken), which is also same as the actual symmetry degree of the improved brake. Table 8 lists the two types of gear mechanisms: a normal gear mechanism and an intermittent gear mechanism. Similar to the previous example, we analyze the factors that influence the symmetry degree and establish the single-factor evaluation matrix based on the characteristics of the structure, given in Table 8. Based on the weight vectors of the factors proposed in the experiment, the comprehensive evaluation of the normal gear mechanism was calculated as B = (b,b 2,b 3 ) = (0.8236,0.427,0.0337); the evaluation result yields v (structure symmetry). The comprehensive evaluation of the intermittent gear mechanism was calculated as B = (b,b 2,b 3 ) = (0.3566,0.4763,0.67); the evaluation result yields v 2 (structure symmetry broken). The evaluation results are in accordance with the actual symmetry degree of the two types of gear mechanisms. 7. CONCLUSION Strict, broken, and weak structure symmetries are the three types of mechanical structure symmetries which play important roles in mechanical systems. They can be distinguished using symmetry degree. To determine the type of symmetry of mechanical structures, we proposed a FCE method to evaluate the symmetry degree of mechanical symmetric structures. There are five steps in the proposed method. First, the set of fac- Transactions of the Canadian Society for Mechanical Engineering, Vol. 4, No. 3,

16 tors is established based on the specific attributes of the four symmetry elements. Second, the evaluation set is established based on the three types of structure symmetry. Third, the value of the single-factor evaluation matrix is set based on the characteristics of the symmetric structure. Fourth, we determine the influences of the factors and set up the weight vectors for the factors. Finally, we proceed with the comprehensive evaluation and obtain the evaluation result. Based on the evaluation result, the symmetry type of the evaluation object can be determined. We proposed basic qualitative descriptions of the evaluation in the single-factor evaluation matrix and designed an experiment to determine the weight vectors. Two instances were used for verifying the proposed evaluation method. The experiment and the instance application both show that the proposed evaluation method can determine the symmetry degree of symmetric structures precisely and effectively. ACKNOWLEDGEMENTS This work was supported by the National Science Foundation of China (No ) and the K. C. Wong Magna Fund in Ningbo University. REFERENCES. Allard, P. and Tabin, C.J., Achieving bilateral symmetry during vertebrate limb development, Seminars in Cell and Developmental Biology, Vol. 20, No. 4, pp , Brading, K.A., Which symmetry? Noether, Weyl, and conservation of electric charge, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, Vol. 33, No., pp. 3 22, Fernandez, E., Symmetry: Key to nature and natural philosophy, Metascience, Vol. 3, No. 3, pp , Kuang, Y.B., Zheng, Y.Q. and Astrom, K., Partial symmetry in polynomial systems and its applications in computer vision, in Proceedings 204 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Columbus, USA, June, pp , Barrenscheen J., Die systematische Ausnutzung von Symmetrieeigenschaften beim Konstruieren, Inst. fur Konstruktionslehre, Maschinen-und Feinwerkelemente, Technische University, Wu, F.B., Bas, G. C., van Schie, Keymer, J.E. and Dekker, C., Symmetry and scale orient Min protein patterns in shaped bacterial sculptures, Nature Nanotechnology, Vol. 0, No. 8, pp. 8, Hargittai, I., Symmetry 2: Unifying Human Understanding, Elsevier, New York, Feng, P.E., Ma, Z.Y. and Qiu, Q.Y., Research on symmetry ontology: from natural science to engineering science, Progress in Nature Science, Vol. 8, No. 2, pp , Ma, Z.Y, Qiu, Q.Y. and Feng, P.E., Concept system and application method of mechanical symmetry, Journal of Zhejiang University (Engineering Edition), Vol. 44, No. 2, pp , Ma, Z.Y., Research on concept system of translation symmetry in mechanical systems, Advanced Materials Research, Vol. 42, pp , Shao, J.P., Feng, Y. and Liu, Y.Q., Automatic identification of part rotational symmetry for dfa evaluation, Computer Integrated Manufacturing Systems, Vol. 6, No. 6, pp , Zheng, Y., Zhang, L.X. and Xiao, T.Y., Symmetry detection algorithm in design for assembly, Journal of Tsinghua University (Sci and Tech), Vol. 48, No. 4, pp , Tate, S.J., Symmetry and Shape Analysis for Assembly-oriented CAD, Cranfield University, London, Liu, R.W., Concept and Application of Mechanical Dynamic Symmetry, Master Thesis of Zhejiang University, Hangzhou, Ma, R.X., Yang, J.G., Qin, P.F., et al., New method for part symmetry automated recognition in DFA, Journal of the China Textile University (Engineering Edition), Vol. 24, No., pp , Lee, S. and Liu, Y., Curved glide-reflection symmetry detection, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 34, No. 2, pp , Transactions of the Canadian Society for Mechanical Engineering, Vol. 4, No. 3, 207

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